# (Non-)Formality of A-infinity algebra implies derived (non-)equivalence?

Take an unital differential graded (dg) $k$-algebra $A$, we can regard it as $A_\infty$-algebra with $m_1$ as differential and $m_2$ as algebra multiplication, and $m_n=0$ or all $n\geq 0$. Take a dg $A$-module $M$, then we can form the $A_\infty$-algebra $B=End_A(M)$.

We say $B$ is formal if the homology $H^*(B)$ is quasi-isomorphic to $B$. Apparently, formality of $B$ implies derived (dg) equivalence, i.e. equivalence between the derived dg categories $D(A)$ and $D(B)$, and hence with $D(H^*(B))$. Is this true? Where can I find an exact reference which states anything like this? I have looked through Professor Keller's note "Introduction to $A_\infty$-algebra and modules", but doesn't seems to see anything like this.

Moreover, is the converse statement true? i.e. if $B$ is not formal, then there is no derived equivalence between $D(A)$ and $D(H^*(B))$.?

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Interesting question, Aaron! Without having checked it out myself, I think that you may be able to find such a statement (if it's true) in Loday and Vallete's "Algebraic operads". An online copy is here: math.unice.fr/~brunov/Operads.pdf –  Aaron Mazel-Gee May 15 '12 at 2:00
Dag Madsen's PhD thesis "Homological aspects in representation theory" seems related. –  Julian Kuelshammer Nov 9 '12 at 12:54

A partial answer to your question: considering the case of Koszul dual algebras and their deformation quantization w.r.t. quadratic Poisson structures it is possible to find the derived equivalence you state considering suitable triangulated subcategories of the $A_{\infty}$ derived categories (not the dg ones). It is possible to state also quite general results (always related to subcategories in the derived categories and using Koszul duality) in the classical case. Please check

http://arxiv.org/abs/0710.5492 (classical dg cases)

http://arxiv.org/abs/1206.2846 ($A_{\infty}$ and quantized cases)

Important: the above derived equivalences are proved by using bimodules.

EDIT The above references study derived equivalences (in the dg and $A_\infty$-cases) of categories of modules over 2 algebras $A$, $B$, s.t. there exists an $A$-$B$-bimodule $M$ which realizes the equivalences. In the $A_\infty$-case the $A_\infty$-bimodule $M$ is s.t. the $A_\infty$-actions $A \rightarrow End_B(M)$ and $B \rightarrow End_A(M)$ are quasi-isomorphisms. These are known as the Keller conditions. They are the crucial assumptions to move forward. As $A$ and $B$ have trivial differentials in the example under consideration, then $A\simeq H(End_B(M))$ and $B\simeq H(End_A(M))$.

As you can see this is a suitable $A_\infty$-Koszul duality theory for the algebras $A$ and $B$. The setting is not the one that you describe in your question (yours is based on a single algebra and module!), but it is quite frequent in applications. For the theory in the dg case, please check Rickard's paper